Just a note: when you differentiate the function once and set and solve for x, you still need to test for the value of the second derivative because you don't know whether that point is a max, min, or neither.

b)there is no minimum in the given interval of because when f'(x)=0 that point is a maximum on the original graph. correct?

Mr F says: There is no maximum at the value of x such that f'(x) = 0 (that is, x = 3pi/4).

c) infection points occur when so that is when and

are these correct? Mr F says: They are the correct solutions to f''(x) = 0. The fact that they do or do not give points of inflection requires justification. Note that x = 3pi/4 is a solution to f'(x) = 0. It can't be a point of inflection AND a maximum turning point (as you say in Q2) .... The resolution of this is left for you to think about for the moment.

so when i first differentiate i get which is the same as then when i differentiate again i get which equals 1 and when . how do i justify that without saying i looked at my gragh?

b)there is no minimum in the given interval of because when f'(x)=0 that point is a maximum on the original graph. correct?

no ... f'(x) = 0 is not the sole indicator that an extrema exists there. you need to perform the 1st or 2nd derivative test to confirm it. in this case, you have endpoint extrema.

many students get confused by this true statement ... if f has an extreme value at x, then f'(x) = 0 or f'(x) is undefined. they want to turn it around and believe that the converse is always true ... it's not.

c) infection points occur when so that is when and

are these correct?

yes, but you need to justify that claim. an inflection point occurs where f''(x) = 0 and f''(x) changes sign.

thank you so much, if i could click the thank you button more than once, i would. i have been trying so many different things. so the min is because that is the endpoint and in a closed interval you have to check the end points. i remember my teacher telling us this a loong time ago, i can't believe i forgot.

but for the points of inflection i am not sure i completely understand. the f''(x) graph i am looking at, it goes from positive y values to negative y values at and then it goes from negative y vlaues to positive y values at . isn't that a sign change?